Parametric tilings and scattered data approximation Michael Floater. International journal of Shape Modeling 4 (1998) pg. 165-187. This paper extends the work of his '97 paper to polygons (as opposed to triangles). It also relates this work to the Harmonic map of Eck, et al . Proofs in paper

• If your mesh has > 3 sided polygons, the convex combination approach still works.
• The shape preserving weights, introduced in the 97 paper, will reproduce all tilings, i.e., given a mesh that's already in the plane, you'll get out the same mesh.
• Choosing weights by the least squares approach (putting weights on the edges) will not always reproduce a tiling.
• The harmonic map introduced by Eck is, indeed, a harmonic map. As long as the weights are well-defined then it will reproduce tilings. Hence, it ends up being similar to the shape preserving approach, in these cases. [Remember that the harmonic map does not make any guarantees about folding.]